Editing Counterexample (section)

==In mathematics==
In mathematics, counterexamples are often used to prove the boundaries of possible theorems. By using counterexamples to show that certain conjectures are false, mathematical researchers can then avoid going down blind alleys and learn to modify conjectures to produce provable theorems. It is sometimes said that mathematical development consists primarily in finding (and proving) theorems and counterexamples.<ref>{{Cite web|url=https://www.cut-the-knot.org/WhatIs/WhatIsCounterexample.shtml|title=What Is Counterexample?|website=www.cut-the-knot.org|access-date=2019-11-28}}</ref>

===Rectangle example===
Suppose that a mathematician is studying [[geometry]] and [[shape]]s, and she wishes to prove certain theorems about them. She [[conjecture]]s that "All [[rectangles]] are [[Square (geometry)|squares]]", and she is interested in knowing whether this statement is true or false. 

In this case, she can either attempt to [[Mathematical proof|prove]] the truth of the statement using [[deductive reasoning]], or she can attempt to find a counterexample of the statement if she suspects it to be false. In the latter case, a counterexample would be a rectangle that is not a square, such as a rectangle with two sides of length 5 and two sides of length 7. However, despite having found rectangles that were not squares, all the rectangles she did find had four sides. She then makes the new conjecture "All rectangles have four sides". This is logically weaker than her original conjecture, since every square has four sides, but not every four-sided shape is a square.

The above example explained — in a simplified way — how a mathematician might weaken her conjecture in the face of counterexamples, but counterexamples can also be used to demonstrate the necessity of certain assumptions and [[hypothesis]]. For example, suppose that after a while, the mathematician above settled on the new conjecture "All shapes that are rectangles and have four sides of equal length are squares". This conjecture has two parts to the hypothesis: the shape must be 'a rectangle' and must have  'four sides of equal length'. The mathematician then would like to know if she can remove either assumption, and still maintain the truth of her conjecture. This means that she needs to check the truth of the following two statements:

# "All shapes that are rectangles are squares."
# "All shapes that have four sides of equal length are squares".

A counterexample to (1) was already given above, and a counterexample to (2) is a non-square [[rhombus]]. Thus, the mathematician now knows that each assumption by itself is insufficient.

===Other mathematical examples===
{{See also|Counterexamples in topology|Minimal counterexample}}

A counterexample to the statement "all [[prime number]]s are [[Parity (mathematics)|odd numbers]]" is the number 2, as it is a prime number but is not an odd number.<ref name=":0" /> Neither of the numbers 7 or 10 is a counterexample, as neither of them are enough to contradict the statement. In this example, 2 is in fact the only possible counterexample to the statement, even though that alone is enough to contradict the statement. In a similar manner, the statement "All [[natural number]]s are either [[Prime number|prime]] or [[Composite number|composite]]" has the number 1 as a counterexample, as 1 is neither prime nor composite.

[[Euler's sum of powers conjecture]] was disproved by counterexample. It asserted that at least ''n'' ''n''<sup>th</sup> powers were necessary to sum to another ''n''<sup>th</sup> power. This conjecture was disproved in 1966,<ref>{{cite journal|last=Lander, Parkin|year=1966|title=Counterexample to Euler's conjecture on sums of like powers|journal=Bulletin of the American Mathematical Society|publisher=Americal Mathematical Society|volume=72|issue=6|page=1079|issn=0273-0979|url=https://www.ams.org/journals/bull/1966-72-06/S0002-9904-1966-11654-3/S0002-9904-1966-11654-3.pdf|access-date=2 August 2018|doi=10.1090/s0002-9904-1966-11654-3|doi-access=free}}</ref> with a counterexample involving ''n''&nbsp;=&nbsp;5; other ''n''&nbsp;=&nbsp;5 counterexamples are now known, as well as some ''n''&nbsp;=&nbsp;4 counterexamples.<ref>{{Cite journal|last=Elkies|first=Noam|date=October 1988|title=On A4 + B4 + C4 = D4|url=https://www.ams.org/journals/mcom/1988-51-184/S0025-5718-1988-0930224-9/S0025-5718-1988-0930224-9.pdf|journal=Mathematics of Computation|volume=51|issue=184|pages=825–835}}</ref>

[[Witsenhausen's counterexample]] shows that it is not always true (for [[control theory|control problems]]) that a quadratic [[loss function]] and a linear equation of evolution of the [[state variable]] imply optimal control laws that are linear.

All [[Euclidean plane isometries]] are mappings that preserve [[area]], but the [[converse (logic)|converse]] is false as shown by counterexamples [[shear mapping]] and [[squeeze mapping]].

Other examples include the disproofs of the [[Seifert conjecture]], the [[Pólya conjecture]], the conjecture of [[Hilbert's fourteenth problem]], [[Tait's conjecture]], and the [[Ganea conjecture]].